reserve y for set,
  x,a for Real,
  n for Element of NAT,
  Z for open Subset of REAL,
  f,f1,f2 for PartFunc of REAL,REAL;

theorem Th1:
  a>0 implies exp_R.(x*log(number_e,a))=a #R x
proof
  assume
A1: a>0;
  number_e<>1 by TAYLOR_1:11;
  then exp_R.(x*log(number_e,a)) =exp_R.(log(number_e,a to_power x)) by A1,
POWER:55,TAYLOR_1:11
    .=exp_R.(log(number_e,a #R x)) by A1,POWER:def 2
    .=a #R x by A1,PREPOWER:81,TAYLOR_1:15;
  hence thesis;
end;
